U.S. patent number 7,443,177 [Application Number 11/421,323] was granted by the patent office on 2008-10-28 for characterization of conductor by alternating current potential-drop method with a four-point probe.
This patent grant is currently assigned to Iowa State University Research Foundation, Inc.. Invention is credited to Nicola Bowler.
United States Patent |
7,443,177 |
Bowler |
October 28, 2008 |
Characterization of conductor by alternating current potential-drop
method with a four-point probe
Abstract
A method of determining material parameters associated with a
conductor using four points includes injecting and extracting
alternating current into the plate using current-carrying wires
operatively connected to two of the four points, measuring
potential drop between the remaining two of the four points, and
calculating the material parameters. The conductor can be of a
homogenous material, a stratified material, or other type of
material. The conductor can have any number of geometries,
including that of a plate, a cylinder, a tube, a stratified
cylinder or other shape.
Inventors: |
Bowler; Nicola (Ames, IA) |
Assignee: |
Iowa State University Research
Foundation, Inc. (Ames, IA)
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Family
ID: |
39874312 |
Appl.
No.: |
11/421,323 |
Filed: |
May 31, 2006 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
Issue Date |
|
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60686061 |
May 31, 2005 |
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Current U.S.
Class: |
324/715; 324/691;
324/713 |
Current CPC
Class: |
G01R
27/02 (20130101) |
Current International
Class: |
G01R
27/08 (20060101) |
Field of
Search: |
;324/715,691,713 |
References Cited
[Referenced By]
U.S. Patent Documents
Foreign Patent Documents
Other References
Bowler, Nicola et al. "Model-Based Characterization of Homogeneous
Metal Plates by Four-Point Alternating Current Potential Drop
Measurements" IEEE Transactions on Magnetics, vol. 41, No. 6, Jun.
2005, pp. 2102-2110. cited by other .
Reynolds, J. M., 1997 Introduction to Applied and Environmental
Geophysics Chichester: Wiley, pp. 426-467. cited by other .
M. Yamashita and M. Agu, "Geometrical Correction factor for
semiconductor resistivity measurements by four-point probe method",
Japanese J. Appl. Phys., vol 23, No. 11, pp. 1499-1504 (1984).
cited by other .
Dodd, C.V. et al. "Measurement of Small Magnetic Permeability
Changes by Eddy Current Techniques", Materials Evaluation 29(10),
Oct. 1971, pp. 217-221. cited by other .
Bowler, Nicola "Part 3. Conductivity Testing" Nondestructive
Testing Handbook, vol. 5 Electromagnet Testing, 2004 3rd ed.,
Chapter 13, Parts 3 and 5, pp. 329-336; pp. 342-343. cited by other
.
Bowler, Nicola et al. "Approximate Theory of Four-Point Alternating
Current Potential Drop on a Flat Metal Surface"; 2005 Meas. Sci.
Technol. 16 2193-200. cited by other .
Mitrofanov V A, "Problems of the Theory of the Electric Potential
Method of Nondestructive Inspect Alternating Current" 1998 Russ. J.
Nondestr. Test. 34 183-9. cited by other .
Bowler, Nicola, "Theory of Four-Point Alternating Current Potential
Drop Measurements on a Metal Half-space"; Journal of Physics D:
Applied Physics 39 (2006) pp. 584-589. cited by other .
Bowler, N. "Frequency-Dependence of Relative Permeability in Steel"
Review of Quantative Nondestructive Eval., vol. 25, 2006; pp.
1269-1276. cited by other .
Bowler, Nicola et al. "Electrical Conductivity Measurement Using
Broadband Eddy-Current and Four-Point Methods", 2005 Center for
Nondestructive Evaluation, pp. 1-16. cited by other .
Huang, Yongqiang "Alternating Current Potential Drop and Eddy
Current Methods for Nondestructive Evaluation of Case Depth", 2004,
Iowa State University, Ames, IA, pp. 1-135. cited by other.
|
Primary Examiner: Gutierrez; Diego
Assistant Examiner: Zhu; John
Attorney, Agent or Firm: McKee, Voorhees & Sease,
P.L.C.
Parent Case Text
PRIORITY STATEMENT
This application claims priority to U.S. Provisional Patent
Application No. 60/686,061, filed May 31, 2005, herein incorporated
by reference in its entirety.
Claims
What is claimed is:
1. A method of determining material parameters associated with a
conductor using four points, comprising: injecting and extracting
alternating current into the conductor using current-carrying wires
operatively connected to two of the four points at each of a
plurality of frequencies; measuring potential drop between the
remaining two of the four points at each of the plurality of
frequencies; and calculating conductivity or thickness of the
conductor using the potential drop and the alternating current
amplitude from a first subset of the plurality of frequencies;
determining magnetic permeability for the conductor from a second
subset of the plurality of frequencies, the second subset of the
plurality of frequencies including frequencies higher than
frequencies in the first subset.
2. The method of claim 1 wherein the conductor is a plate.
3. The method of claim 2 wherein the thickness is plate
thickness.
4. The method of claim 1 wherein the conductor is a stratified
conductor.
5. The method of claim 1 wherein the conductor is a conductive
surface treatment.
6. The method of claim 1 wherein the conductor is a coating.
7. The method of claim 1 wherein the conductor comprises a ferrous
metal.
8. The method of claim 1 wherein the four points are co-linear.
9. The method of claim 1 wherein the four points are in a
rectangular configuration.
10. The method of claim 1 wherein the four points are randomly
placed.
11. The method of claim 1 wherein the conductor comprises a metal
half-space.
12. The method of claim 1 wherein the step of calculating includes
applying an analytical expression.
13. The method of claim 1 wherein the conductor is of cylindrical
shape.
14. The method of claim 1 wherein the conductor is a tube.
15. The method of claim 1 wherein the conductor is a stratified
cylinder.
16. A method of determining material parameters associated with a
conductor by using a four points probe, the method comprising:
injecting and extracting alternating current into the conductor
using current-carrying wires operatively connected to two of the
four points of the four point probe at each of a plurality of
frequencies; measuring potential drop, V, between the remaining two
(p, q) of the four points of the four point probe at each of the
plurality of frequencies; applying an analytical expression
relating the voltage, V, measured between points p and q at each of
the plurality of frequencies, the electromagnetic skin depth of the
conductor and the amplitude of the alternating current to determine
the magnetic permeability based on a first subset of the plurality
of frequencies and conductivity of the conductor based on a second
subset of the plurality of frequencies, the first subset including
frequencies higher than the frequencies in the second subset.
17. The method of claim 16 wherein the conductor is a homogenous
metal.
18. The method of claim 17 wherein the conductor is a plate.
19. The method of claim 16 wherein the conductor comprises a
ferrous material.
20. A method of determining material parameters associated with a
conductor using four points, comprising: injecting and extracting a
time-varying current into the conductor using current-carrying
wires operatively connected to two of the four points; measuring
potential drop between the remaining two of the four points using a
measurement circuit; determining the material parameters of
conductivity, thickness, and magnetic permeability from the
potential drop, characteristics of the time-varying current, and an
analytic model of an electromagnetic field in the conductor; and
wherein the analytic model provides for defining a complex voltage
for the potential drop with a first contribution to the complex
voltage due to the time-varying current into the conductor and a
second contribution to the complex voltage due to induction in a
loop of the measurement circuit, evaluating both the first
contribution and the second contribution to thereby decouple the
conductivity from the magnetic permeability and provide for
calculating both the conductivity and the magnetic permeability.
Description
BACKGROUND OF THE INVENTION
The present invention relates generally to the characterization of
conductive objects or surfaces, such as, without limitation metal
plates, rods, coatings using non-destructive evaluation techniques
and in particular using a four-point probe. Measurements of
electrical conductivity (or resistivity) are useful in metal
sorting, alloy identification, heat-treatment monitoring of
aluminum alloys, and detection of flaws which are manifest as a
change in the material conductivity, such as thermal damage in
aircraft structures. Prior art methods include existing four-point
methods for measuring metal conductivity or thickness that use
direct current or the use of standard eddy-current instruments for
measuring metal conductivity. These types of approaches have
limited utility. For example, where direct current is used only a
limited number of material properties can be accurately determined.
In addition, there are problems with use of too much current, ionic
deposition problems, the need for geometric correction factors, and
a host of other problems. Standard eddy-current instruments for
measuring metal conductivity cannot be used in the case of ferrous
metals.
BRIEF SUMMARY OF THE INVENTION
Therefore it is a primary, object, feature, or advantage of the
present invention to improve upon the state of the art.
It is a further object, feature, or advantage of the present
invention to provide for using four contact points on a conductive
object to determine characteristic information about the conductive
object such as dimensions, electrical conductivity, and magnetic
permeability.
Yet another object, feature, or advantage of the present invention
is to provide for using four contact points on a metal plate to
determine characteristic information about a metal plate such as
thickness, electrical conductivity, and magnetic permeability.
A further object, feature, or advantage of the present invention is
to provide for using four contact points on a metal rod to
determine characteristic information about the metal rod.
Yet a further object, feature, or advantage of the present
invention is to provide for using four contact points on a
conductive surface to determine characteristic information about
the conductive surface, including where the conductive surface is a
coating.
Another object feature, or advantage of the present invention is to
provide for a method of determining both electrical conductivity
and magnetic permeability.
Yet another object, feature, or advantage of the present invention
is to provide a method that uses a multiple frequency approach
rather than a direct current or single-low frequency approach.
A still further object, feature, or advantage of the present
invention is to provide for a four contact point method to
determine characteristics of a conductive plate comprised of
ferrous metals.
Another object, feature, or advantage of the present invention is
to provide a method for accurate measurement of small changes
and/or differences in magnetic permeability.
Yet another object, feature, or advantage of the present invention
is to provide a method that can assist in determining the
processing history of metals such as stainless steels by accurate
measurement of small changes in magnetic permeability when the
value is close to unity.
A further object, feature, or advantage of the present invention is
to provide a method of measuring electrical conductivity that is
accurate and consistent.
A still further object, feature, or advantage of the present
invention is to provide a method of measuring electrical
conductivity that becomes more accurate as conductivity
decreases.
Another object, feature, or advantage of the present invention is
to provide a method for measuring magnetic permeability which is
simple and low cost.
Yet another object, feature, or advantage of the present invention
is to provide a method for measuring electrical conductivity which
does not rely on time-consuming prior calibrations.
A further object, feature, or advantage of the present invention is
to provide for an accurate, portable conductivity measurement
system.
One or more of these and/or other objects, features, or advantages
of the present invention will become apparent from the
specification and claims that follow.
According to one aspect of the present invention, a method of
determining material parameters associated with a conductor using
four points is provided. The method includes injecting and
extracting alternating current into the conductor using
current-carrying wires operatively connected to two of the four
points, measuring potential drop between the remaining two of the
four points, and calculating the material parameters of the
conductor using the potential drop and the alternating current. The
material parameters which can be measured include, magnetic
permeability, electrical conductivity (or resistivity), geometry
parameters (such as plate thickness, when the conductor is a
plate). The conductor can be, for example, a metal plate, a
stratified conductor, a conductive surface treatment, a coating, a
ferrous metal, a metal half space, metal rod or tube. The four
points may be co-linear, may be oriented in a rectangular
configuration or randomly or arbitrarily placed.
According to another aspect of the present invention a method is
provided for determining material parameters associated with a
conductor by using a four points probe. The method includes
injecting and extracting alternating current into the conductor
using current-carrying wires operatively connected to two of the
four points of the four point probe, measuring potential drop, V,
between the remaining two (p, q) of the four points of the four
point probe, and applying an analytical expression relating the
voltage, V, measured between points p and q, the electromagnetic
skin depth of the conductor and the amplitude of the alternating
current to determine the magnetic permeability and conductivity of
the conductor.
Generally, the method relies on electrical contact to a conductor
by means of four points. Two points facilitate injection and
extraction of alternating current into/out of the plate means of
current-carrying wires. The other two facilitate measurement of
potential drop (voltage) between points on the surface of the
conductor by a high-impedance voltmeter. The measured voltage can
be used to infer characteristic information about the conductor and
properties such as electrical conductivity, and magnetic
permeability.
The present invention uses alternating current at a number of fixed
frequencies. This permits a greater number of material parameters
to be determined than if direct current is used, including accurate
determination of magnetic permeability. Generally, the present
invention needs less current, avoids ionic deposition problems,
requires fewer geometric correction factors and may give a larger
voltage signal than prior art methods.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a schematic showing a path of integration.
FIG. 2 is a graph showing ACPD measurements on a brass plate
compared with theory.
FIG. 3 is a graph showing calculated values of as a function of
frequency and plate thickness.
FIG. 4 is a graph showing calculated values of Im() as a function
of frequency and perpendicular length of the pick-up wire, l.
FIG. 5 is a schematic illustrating a four point probe in contact
with a conductive half-space. The path of integration, C (- - - -),
may occupy any plane of constant .gamma.. Here the plane y=0 is
shown. l is the dimension of the pickup circuit perpendicular to
the conductor surface.
FIG. 6 is a graph illustrating a dimensionless pickup voltage,
.pi..sigma..nu..sup.LSS/I, as a function of dimensionless
frequency, .omega..mu..sigma.S.sup.2, in the case of a co-linear,
symmetric probe, for q/S=1/3(0.333), 3/5(0.600), 5/7(0.714),
7/9(0.778) and 9/11(0.818).
FIG. 7 is a graph illustrating dimensionless pickup voltage,
.pi..sigma..sup.RS/I, as a function of dimensionless frequency,
.omega..mu..sigma.S.sup.2, in the case of a rectangular probe, for
c/S=2, 2/3 (0.667), (0.400), 2/7(0.286), 2/9(0.222) and
2/11(0.182).
FIG. 8 is a graph illustrating the imaginary part of the
dimensionless pickup voltage, .pi..sigma..nu..sup.LSS/I, as a
function of dimensionless frequency, .omega..mu..sigma.S.sup.2, in
the case of a co-linear, symmetric probe, with q/S=1/3. Curves are
plotted for various values of the parameter l, the height of the
pickup loop above the conductor surface (FIG. 5). The area of the
pickup loop, and hence the inductance of the loop, is proportional
l.
FIG. 9 is a graph illustrating impedance (V/I) measured by a
co-linear, four-point probe (table 3) in contact with an aluminum
block (table 4), compared with theory expressed in equation (B.30),
as a function of frequency.
FIG. 10 is a graph illustrating impedance (V/I) measured by a
rectangular, four-point probe (table 3) in contact with an aluminum
block (table 4), compared with theory expressed in equation (B.30),
as a function of frequency.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
The present invention provides a method of quantifying both
electrical conductivity and magnetic permeability. The methodology
can be applied to objects of varying geometries, including, but not
limited to plates, cylindrical rods, and other geometries. In
addition, the methodology can be used for characterizing surface
treatments or other forms of layered structures in metals. In other
words, the object being characterized need not be homogenous. Thus,
the present invention provides a flexible methodology for
quantifying or characterizing electrical conductivity and magnetic
permeability for a variety of materials in a variety of
environments or situations.
Measurements of voltage are taken at a number of specific
frequencies. The low-frequency voltage values are entered into a
theoretical formula to calculate either the metal conductivity or
thickness. One of these must be known independently in order to
calculate the other. Higher-frequency voltage values are entered
into a theoretical formula (along with the conductivity and plate
thickness) to calculate the magnetic permeability. On samples
tested, conductivity measurements are accurate to within 1% of
values obtained by an independent (eddy-current) method. The
MIZ-21A eddy-current instrument measures conductivity to at best 1%
and possibly only 20% accuracy for low conductivity metals such as
stainless steel. In order to measure magnetic permeability by an
eddy-current method, low-frequency instrumentation is required
because the conductivity and permeability remain coupled in their
behavior until low frequencies are reached. In the proposed method,
the conductivity becomes decoupled from the permeability at
significantly higher frequency, making it possible to measure both
parameters easily using the same technology. Very small variations
in magnetic permeability (.+-.0.01) are observable for low
permeability metals.
The method relies on a theoretical formula for interpretation of
the measurement data and deduction of the parameters of the metal
plate. This theoretical formula has not previously been derived in
the case of alternating current.
The alternating current potential drop (ACPD) method measures the
voltage, , between two pick-up points on the surface of a
conductor. For the configuration shown in FIG. 1,
.intg..times..times.d .times.d.times. ##EQU00001## where C is a
closed loop [1]. .epsilon. is the rate of change of magnetic flux
within the loop.
In direct current potential drop measurements there is no induction
effect in the measurement circuit (.epsilon.=0) since the current
does not vary with time and in this case the measured potential
drop is almost exclusively due to the conductor. In ACPD
measurements, the contribution to from the conductor dominates when
the frequency is sufficiently low, since the inductive contribution
from the measurement circuit, i.omega.L, is proportional to
frequency .omega.. At sufficiently high frequency the inductive
term dominates. In this work, both contributions to fare evaluated.
The far-field approximation for E is used in calculating . This
approximation gives accurate results when pick-up points at (p, y,
0) and (q, y, 0) are sufficiently far from the source points at
(.+-.S, 0, 0), in practice a few electromagnetic skin depths
(.delta.) in the conductor and the conductor is somewhat thinner
than the probe spacing, as shown for direct current potential drop
measurements [10, 11].
Electric Field
For the configuration shown in FIG. 1, the electric field can be
obtained by superposition of fields separately associated with the
two current-carrying wires: E.sup.T(r)=E(r.sub.+)-E(r.sub.-), (A.2)
where r.sub..+-.= {square root over
((x.+-.S).sup.2+y.sup.2+z.sup.2)}. In the following sections the
far-field form of E is determined in the region of the pick-up
circuit (air) and in the metal plate for a single current-carrying
wire located on the axis of a cylindrical co-ordinate system. Probe
Region
For a single wire passing current I into, or out of, a conductive
plate, there are two contributions to the electric field in air.
One is from the current flowing in the wire, E.sup.W, and the other
is from the current density in the plate. In the far-field regime,
for the closed loop C, only E.sup.W is important. Assuming that the
wire is perpendicular to the surface of the plate and that the
current has time-dependence e.sup.-i.omega.t, the integral form of
Ampere's Law and then Faraday's Law yields
.function..rho..times.I.omega..times..times..mu..times..times..pi..times.-
.times..times..rho..times..rho..fwdarw..infin..ltoreq..times.
##EQU00002## where .rho. is the radial co-ordinate of a cylindrical
system centered on the wire and E.sup.W has the same direction as
the current density in the wire, J={circumflex over (z)}J.sub.z.
Plate
An expression for the electric field in the conductive plate is
obtained in a manner similar to that given in reference [2] for a
conductive half-space. For a current source oriented perpendicular
to the surface of the plate, only the transverse magnetic (TM)
potential, .psi.'', is required to fully describe the electric
field:
E(r)=-i.omega..mu..gradient..times..gradient..times.{circumflex
over (z)}.PSI.''(r) (A.4) Define a Modified TM Potential
.PSI.=.gradient..sub.z.sup.2.psi.'', (A.5) where
.gradient..sub.z.ident..gradient.-{circumflex over
(z)}(.differential./.differential.z) is the transverse differential
operator. For a plate infinite in x and y, occupying
z.epsilon.[0,T], the governing equation is
(.gradient..sup.2+k.sup.2).PSI.(r)=0,0.ltoreq.z.ltoreq.T, (A.6)
where k.sup.2=i.omega..mu..sigma. with .mu. and .sigma. being the
permeability and conductivity of the plate, respectively. In the
plate, only the horizontal component of the electric field,
E.sub..rho., contributes to V. It is not convenient to express
E.sub..rho. in terms of .PSI.. Rather, E.sub..rho. will be obtained
from the following equation by means of relationship (A.5).
.rho..function.I.times..times..omega..mu..times..differential..times..psi-
.''.function..differential..rho..times..differential..times.
##EQU00003## where .rho. and z are co-ordinates of the cylindrical
system. Equation (A.6) is solved for T subject to boundary
conditions
.PSI..function..rho..function..rho..times..times..times..times..function.-
.rho..pi..function..times..times..rho..ltoreq..rho.>.times.
##EQU00004## and .PSI.(.rho.,T)=0. (A.9) These derive from the fact
that, at the surface of the plate, the normal component of current
density is continuous-zero everywhere apart from at the point of
contact with the current carrying wire, radius .alpha.. Applying
the zero-order Hankel transform to solve (A.6) and taking the limit
.alpha..fwdarw..about.0 yields
.PSI..function..rho..times..pi..times..times..times..intg..infin..times.e-
.gamma..times..times..function.e.times..gamma..function.e.times..gamma..ti-
mes..times..times..function..kappa..rho..times..kappa..times..times.d.kapp-
a..times. ##EQU00005## where y.sup.2=k.sup.2-k.sup.2. If
T.fwdarw..infin., the term in square brackets tends to unity and
the resulting integral is identical to that obtained for a
half-space conductor [2].
It is possible to evaluate the integral (A.10) analytically by
expanding the term in the denominator as a binomial series [4,
3.6.10]:
e.times..gamma..times..times.e.times..gamma..times..times.e.times..gamma.-
.times..times.e.times..gamma..times..times.e.times..gamma..times..times..i-
nfin..times..times.e.times..times..times..times..gamma..times..times..time-
s. ##EQU00006##
Multiplying the right-hand side of (A.11) by the factor
e.sup.-yz[1-e.sup.2y(z-T)]and substituting the result into (A.10)
yields
.PSI..function..rho..times..pi..times..times..times..infin..times..times.-
.intg..infin..times.e.gamma..times..times..times..times..times..times.e.ga-
mma..function..times..times..times..function..kappa..rho..times..kappa..ti-
mes..times.d.kappa..times. ##EQU00007##
where the order of summation and integration has been reversed. The
first term in braces in (A.12), e.sup.-.gamma.z, gives rise to the
result for the TM potential in a half-space conductor. The second
term, -e.sup..gamma.(z-2T), accounts for the primary reflection of
the field from the surface of the plate at z=T. Other terms deal
with multiple reflections between the surfaces of the plate. By
analogy with the result for the half-space conductor, reference
[2], or by multiple use of the analytic result given in reference
[5], result 8.2.23, the terms in (A.12) can be integrated. It is
found that
.PSI..function..rho..times..pi..times..infin..times.I.times..times..funct-
ion..times..times..times..times.I.times..times..times..times..times.eI.tim-
es..times..times..times..function.I.times..times..times..times.I.times..ti-
mes..function..times..times.I.times..times..times..times.'.times.eI.times.-
.times..times..times.'.function.I.times..times..times..times.'.ltoreq..lto-
req..times. ##EQU00008## wherein r.sub.n= {square root over
(.rho..sup.2+(z+2nT).sup.2)} and r'.sub.n= {square root over
(.rho..sup.2+[z-3(n+1)T].sup.2.)}
To obtain E.sub..rho. from .PSI. as given in (A.13) via relations
(A.7) and (A.5) requires some manipulation [2]. The result is
.rho..function.I.times..times..times..times..times..pi..times..times..sig-
ma..rho..times..infin..times..times.eI.times..times..function..times..time-
s..times..times.eI.times..times..times..times.I.times..times..times..times-
..function.I.times..times..function..times..times..times..times.I.times..t-
imes..times..times..times.I.times..times..times..times.eI.times..times..fu-
nction..times..times.eI.times..times..times..times.'I.times..times..times.-
.times.'.function.I.times..times..function..times..times.I.times..times..t-
imes..times.'.times.I.times..times..times..times.'.ltoreq..ltoreq..times.
##EQU00009##
In the far field, the electric field is dominated by terms of the
form e.sup.ikz/.rho. and
.rho..function.I.times..times..times..times..times..pi..times..times..sig-
ma..rho..times..infin..times..times.e.times..times..function..times..times-
..times..times.eI.times..times..function..times..times..rho..fwdarw..infin-
..ltoreq..ltoreq..times. ##EQU00010##
If the far-field current density is integrated over a cylindrical
surface of large radius extending from z=0 to T, the result is
I.left brkt-bot.1+e.sup.ik(2N-1)T.right brkt-bot. for a series
truncated to N terms. This expression tends to I as
N.fwdarw..infin., as it should. If T.fwdarw..infin. the far-field
expression for the electric field in a half-space conductor is
recovered [2]
.rho..function..rho..times.I.times..times..times..times..times..pi..times-
..times..sigma..times..times..rho..times.eI.times..times..times..times..ti-
mes..rho..fwdarw..infin..times..gtoreq..times. ##EQU00011##
This expression was also given in reference [3] in the context of
fatigue crack measurement.
Voltage Calculation
Voltage is now calculated according to Equation (A.1). For the
configuration shown in FIG. 1 the contributions are
.intg..times..function..times..times.d.intg..times..function..times..time-
s.d.intg..times..function..times..times.d.times. ##EQU00012##
with E.sup.T given by (A.2). It is a simple matter to evaluate the
last two terms on the right-hand side of equation (A.17) with
E.sub.z given in equation (A.3). To neatly evaluate the first term
on the right-hand side of (A.17) recognize that, at the surface
defined by z=0, equation (A.15) can be written
.rho..function..rho.I.times..times..times..times..times..pi..times..times-
..sigma..times..times..rho..function..times..infin..times..times.e.times..-
times.I.times..times..times..times..times..times..times..rho..fwdarw..infi-
n..times. ##EQU00013##
Further [4, equation 3.6.10],
.infin..times..times.e.times..times.I.times..times..times..times..times..-
times.e.times..times.I.times..times..times..times. ##EQU00014##
so that
.rho..function..rho.I.times..times..times..times..times..pi..times..times-
..sigma..times..times..rho..times..function.I.times..times..times..times..-
times..rho..fwdarw..infin..times. ##EQU00015##
The final expression for is
.times..pi..function.I.times..times..sigma..times..function.I.times..time-
s..times..times.I.times..times..omega..times..times..mu..times..times..tim-
es..function..times. ##EQU00016##
The first term in equation (A.20) is the contribution from the
conductor and has approximately equal real and imaginary parts. The
contribution from the measurement circuit is imaginary (inductive)
and proportional to the dimension of the circuit perpendicular to
the conductor surface, l. For a typical non-magnetic metal and
l.about.1 mm, the inductive term is practically negligible for
frequencies up to about 10 Hz whereas at 10.sup.4 Hz the terms are
of similar magnitude. The logarithmic term represents the physical
arrangement of the four probe points.
Experiment
ACPD measurements were made as a function of frequency on a brass
plate whose conductivity and dimensions are given in Table 1. The
brass plate was precision ground to remove surface scratches and
mounted on a two-inch thick plastic support plate. Electrical
contact with the brass plate was made via sprung, point contacts,
held perpendicular to the surface of the plate. In this experiment
the four contact points were arranged in a straight line, with a
common midpoint between the two current drive points and the two
pick-up points. The dimensions of the probe are given in Table
1.
TABLE-US-00001 TABLE 1 Experimental parameters. brass plate probe
(FIG. 1) conductivity, .sigma. (MSm.sup.-1) 16.2 .+-. 0.3 S (mm)
38.2 .+-. 0.3 thickness, T (mm) 5.66 .+-. 0.01 p (mm) -9.18 .+-.
0.01 horizontal dimensions (mm) 615 .times. 616 q (mm) 9.18 .+-.
0.01 l (mm) 0.35 (fitted value)
The two current-carrying wires were held perpendicular to the plate
surface for a distance of 16 inches, after which they were twisted
together to reduce the effects of inter-wire capacitance. This
distance was sufficient to remove any effect of motion of the
current wires on the measured voltage. The two pick-up wires were
arranged with the objective of minimizing l, lying as close to the
plate surface as possible. They were twisted together at the
midpoint between the pick-up points.
In the theoretical calculation, two measured values are needed. One
is the current through the plate, the other is the voltage measured
by the pick-up probe. To monitor the current in the plate, a high
precision resistor was connected in series with the drive current
circuit and the voltage across the resistor measured. The
resistance maintains one percent accuracy over the range of
frequency for which it could be measured with an Agilent 4294A
precision impedance analyzer; 40 Hz to 40 kHz. The voltage across
the resistor and that of the pick-up probe were both measured using
a Stanford Research Systems SR830 DSP lock-in amplifier.
In order to make both voltage measurements using the same lock-in
amplifier, a switch was used activated by a control signal from the
auxiliary analog output of the lock-in amplifier. It was necessary
to correct the experimental data for common-mode rejection (CMR)
error in the lock-in amplifier. This systematic error shows itself
in the fact that, when the pickup terminals are reversed, the
measured voltage changes by a few .mu.V. The magnitude of the error
is, therefore, similar to that of the voltage being measured, and a
corrective procedure is essential. The CMR error was eliminated by
taking two sets of measurements, reversing the pick-up terminals
for the second. The two sets were then subtracted and the result
divided by two.
The drive current was produced by a Kepco bipolar operational power
supply/amplifier, model number BOP 20-20M. The sine signal from the
internal function generator of the lock-in amplifier was connected
to the current programming input of the power supply, with the
power supply working as a current drive.
The conductivity of the plate was measured using a MIZ-21A eddy
current instrument. The error quoted in Table 1 is estimated from
the manufacturer's literature and derives from a combination of
inaccuracy in the instrument, inaccuracy in the comparative
standards and probe lift-off error.
In FIG. 2, ACPD measurements are compared with theory. The average
of ten data sets (taken sequentially) is shown. The value of I was
adjusted in the calculation to obtain the best fit to the high
frequency imaginary part of the data, having negligible influence
on the low-frequency data. The value l=0.35 mm appears reasonable
since the pick-up wire is AWG 32 with diameter 0.2 mm. The
agreement between theory and experiment is excellent. There is no
obvious error in the imaginary part of V. The theory overestimates
the low frequency real part of Vby 3%. Applying standard error
analysis to the low frequency limiting expression for V, equation
(A.22), shows that errors in the plate conductivity and in the
relative positions of the probe points combine to give an
experimental error which is also 3%.
Limiting Case: Low Frequency
To take the limit k.fwdarw.0 in equation (A.20), not that
lim.sub.k.fwdarw.0[ikT/sinh(ikT)]=1. Then
.fwdarw..times..pi..times..times..sigma..times..times..times..times..func-
tion..times..fwdarw..times. ##EQU00017## It is seen that at low
frequency the voltage is real, being inversely proportional to the
plate thickness and conductivity. Formula (A.22) is consistent with
one given by Yamashita and Masahiro for four-point DC measurements
on a finite plate [6]. In FIG. 3, is plotted for a number of values
of plate thickness. The inverse dependence of Re() on the plate
thickness at low frequency, predicted by equation (A.22), can be
clearly seen in FIG. 3. Limiting Case: High Frequency
At high frequency the voltage is dominated by the inductive term in
equation (A.20). This term is proportional to l, the length of the
pick-up wire perpendicular to the metal plate. Practically it is
desirable to minimize the contribution of this term by making l as
small as possible. In this way the contribution to due to the
plate, from which useful information may be derived, is not masked
by induction in the measurement circuit. In FIG. 4, the effect on
of varying l is shown. Only Im() is shown since l has no influence
on Re().
Therefore, a four-point method of measuring material parameters has
been disclosed. The simple analytic result, equation (A.20), gives
useful insight into the primary contributors in ACPD measurements.
It is accurate for a flat plate whose edges are several tens of
skin depths from the probe, for a probe whose pick-up points are
several skin depths away from the current drive points and for a
conductor somewhat thinner than the probe point spacing.
Of course the present invention has numerous applications,
including but not limited to use in non-destructive quantification
of metal plates of titanium/nickel alloys such as used in aircraft
engines, metal billets, and other uses where material evaluation or
process monitoring is used. These applications further include
applications where non-destructive quantification of metal
thickness is needed and access is restricted to one side. The
present invention has been used to measure a variety of different
types of conductors including brass, aluminum, stainless steel,
carbon steel and spring steel. The present invention is not to be
limited to the specific disclosure presented herein, as one skilled
in the art having the benefit of this disclosure would appreciate
the far-reaching scope of the present invention.
Metal Half Spaces
The present invention also provides for the use of four-point
alternating current potential drop measurements on a metal
half-space. An analytic expression is derived and used to describe
the complex voltage measured between the pickup points of a
four-point probe, in contact with the surface of a half-space
conductor. The alternating current potential drop (ACPD)
measurements permit depth-dependent information to be obtained
through the phenomenon of the electromagnetic skin effect, in which
the current is confined to flow in a `skin` at the surface of the
conductor, whose depth is approximately inversely proportional to
the square root of the excitation frequency. The ACPD technique
therefore has application in assessing materials whose
electromagnetic parameters vary with depth, for example, in the
case of electrically conductive surface treatments and coatings.
One advantage of ACPD over DCPD is that a lower measuring current
can be applied in order to achieve a given sensitivity [10]
(section 8). This reduces the risk of heating of the specimen and
associated changes in electrical conductivity.
In previous work, Mitrofanov derived an expression for the complex
voltage measured between the pickup points, of a four-point probe,
in contact with the surface of a half-space conductor [12]. The
solution was expressed in terms of an infinite series expansion in
powers of k, where
I.delta..times..times..times..times..delta..omega..times..times..mu..time-
s..times..sigma..times. ##EQU00018##
.delta. being the electromagnetic skin depth in the conductor. In
equation (B.1), .omega.=2.pi.f is the angular frequency of the
injected current and .mu. and .sigma. are the magnetic permeability
and electrical conductivity of the half-space, respectively.
The analytical expression describing the complex voltage measured
between the pickup points of a four-point probe, in contact with
the surface of a half-space conductor, is derived in closed form.
There are two contributions to the measured voltage. One arises
from the potential drop due to electric current flowing in the
conductor. The other arises from induction in the loop of the
pickup circuit. Both terms are obtained by integrating analytic
expressions for the electric field, derived previously [2, 13],
along appropriate paths. It is shown that the closed-form
expression obtained here for the potential drop due to current
flowing in the conductor can be expressed as a power series in k,
giving the same result as that presented in [12]. The contribution
to the measured complex voltage due to inductance in the pickup
circuit was not analyzed in [12].
Theory is compared with experimental data for co-linear and
rectangular arrangements of the four probe points in contact with a
thick aluminum block and very good agreement is obtained in both
cases.
Analysis
The ACPD method measures a complex voltage, V, which has two
contributions: V=v+.epsilon. (B.2)
The first term, v, is the potential drop between the two points on
the plate at which the measurement circuit makes contact with its
surface. The source of v is the current in the plate injected by
the other two points of the four-point probe. At arbitrary
frequency, v is complex. The second contribution, .epsilon., is
proportional to the inductance of the measurement circuit. It
arises form the changing magnetic flux within the loop of the
measurement circuit due to harmonic variation of the applied
current, of the form e.sup.-i.omega.t, .epsilon. is purely
inductive, therefore imaginary. In the static limit of direct
current, only v remains. For the geometry given in FIG. 5,
.intg..times..times.d.times. .times.d.times. ##EQU00019##
Where C is a closed loop in the case where p' and q' coincide, as
happens when the pickup wires are twisted together at their point
of meeting. At low frequency, the measured potential drop is almost
exclusively due to the conductor. In an ACPD measurement on a
conductive plate, the contribution to V from the plate is most
significant at lower frequencies, with the contribution from
.epsilon. becoming larger, and eventually dominant as the frequency
increases. Strictly, the quantities V, .nu., .epsilon. and E are
complex amplitudes. For brevity, the time dependence is not shown
explicitly in equations (B.2) to (B.4) or in the equations that
follow.
For current injected into a half-space conductor by a single wire
held perpendicular to the conductor surface, the components of the
electric field in the conductor are [2]
.rho..function..times..pi..times..times..sigma..times.I.times..times..rho-
..times.eI.times..times..times..times.eI.times..times..times..times.I.time-
s..times..times..times..function.I.times..times..times..times.I.times..tim-
es..times..times..times.I.times..times..times..times.>.times..function.-
.times..pi..times..times..sigma..times..times.eI.times..times..times..time-
s..function.I.times..times..times..times.>.times. ##EQU00020##
in which .rho. and z are the variables of a cylindrical co-ordinate
system centered on the current wire and
r.sup.2=.rho..sup.2+z.sup.2. The electric field in air may be
expressed [13] as E.sup.S=E.sup.W+E.sup.C,.rho.>0,z.ltoreq.0,
(B.7) where
.times..times..pi..times.I.omega..mu..times..times..times..rho..rho.>.-
ltoreq..times. ##EQU00021## and
.times..pi..sigma..times..intg..infin..times..gamma.e.kappa..times..times-
..function..rho..times..function..kappa..rho..times..function..kappa..rho.-
.times..times.d.kappa..ltoreq..times. ##EQU00022## In equation
(B.9), y.sup.2=k.sup.2-k.sup.2 and J.sub.i(x)is the i-th order
Bessel function of the first kind. E.sup.W is the electric field in
air due to the current flowing in the injection wire. E.sup.C is
the electric field in air due to the current flowing in the
half-space conductor.
For a system of two current-carrying wires in contact with the
metal surface at co-ordinates (.+-.S,0,0) as shown in FIG. 5, the
electric field E can be obtained by the superposition of the field
due to a single wire, E.sup.S, whose components are given above:
E(r)=E.sup.S(r.sub.+)-E.sup.S(r.sub.-) with r.sub..+-.= {square
root over ((z.+-.S).sup.2+y.sup.2+z.sup.2)} Calculation of .nu.
Closed form. In general, the line of the pickup points may be
off-set from the line of the current injection points. Let
y=c=constant and then choose the path of the integral in equation
(B.3) such that .nu.=-.intg..sub.p.sup.qE.sub.x(x,c,0)dx (B.11)
Now,
.function..rho..times..rho..function..rho..times..rho..function..rho..tim-
es. ##EQU00023## where .rho..sub..+-.= {square root over
((x.+-.S).sup.2+c.sup.2)}. Combining the above two equations and
making the change of variable X=x.+-.S gives .nu.=-I.sub.++I.sub.-,
(B.13) where
.+-..intg..rho..+-..+-..times..times..rho..function..times..times.d.times-
. ##EQU00024## Putting E.sub..rho..sup.S(X,c,0) from equation (B.5)
into the integrand of equation (B.14) gives
.+-.I.times..times..times..times..times..pi..times..times..sigma..times..-
intg..rho..+-..+-..times..times..times.eI.times..times..times.I.times..tim-
es..function..times.d.times. ##EQU00025## Integration of the first
term in equation (B.15) is straightforward. The second term in
equation (B.15) may be evaluated by making a further change of
variable, .alpha.= {square root over (X.sup.2+c.sup.2)}, and using
the following identity (equation 2.325.2) in [14]):
.intg.e.times.de.function..times. ##EQU00026## in which E.sub.1(z)
is the exponential integral function, defined (equation (5.1.1) in
[15]) as
.function..intg..infin..times.e.times..times.d.times..times.<.pi..time-
s. ##EQU00027## Ultimately, the following expression for .nu. is
obtained:
.times..times..pi..times..times..sigma..function..function..times..functi-
on..function..times..function..times. ##EQU00028## where, as will
be shown subsequently, f.sub.i(x,y) can take several forms. In
exact, closed form,
.function..function..rho.eI.times..times..times..times..rho..rho.I.functi-
on..times..times..rho..function.I.times..times..times..times..rho..times.
##EQU00029## Series Form The result presented in equations (B.18)
and (B.19) can be expressed in terms of a power series in k. In
this way it can be shown that the result is in agreement with that
of an independent calculation [12]. The two relations (equations
(4.2.1) and (5.1.11) in [15]
e.infin..times..times. ##EQU00030##
.function..gamma..times..times..infin..times..times..times..times.<.pi-
..times. ##EQU00031## applied to the exponential and exponential
integral functions in equation (B.19) give
f.sub.exact(.rho.)=-ik[.gamma..sub.e+ln(-ik)-1]+f.sub.series(.rho.)
(B.22) in which y.sub.e=0.577216 . . . is Euler's constant and
.times..function..rho..rho..function..infin..times.I.times..times..times.-
.times..rho..function..times. ##EQU00032## Note that the terms
present in the relation between f.sub.exact and f.sub.series
(equation (B.22)) are independent of .rho.. This means that they
drop out when inserted into equation (B.18). Hence f.sub.series
(equation (B.23)) may be inserted directly into equation (B.19) as
an alternative to f.sub.exact (equation (B.19)). The resulting
series representation for .nu. given by combining equations (B.18)
and (B.23) agrees with that presented in [12]. Special Cases
One commonly-used probe configuration is that in which the four
probe points are arranged along a straight line, with the voltage
pickup points positioned symmetrically about the midpoint between
the current injection points. In the case of this co-linear,
symmetric probe, p=-q and c=0. Equations (B.18) reduces to
.pi..times..times..sigma..function..function..function..times.
##EQU00033##
For a rectangular probe configuration, in which the line between
the current injection points forms one side of the rectangle and
that between the voltage pickup points forms the opposite side,
p=-S and q=S so that
.pi..sigma..function..function..times..function..times.
##EQU00034## In the limit of direct current, k.fwdarw.0 and
.function..rho..times..function..rho..rho..times. ##EQU00035## in
agreement with results presented in [9, 11, 15].
In FIG. 6, the real and imaginary parts of the dimensionless
voltage, .pi..sigma..nu..sup.LS/I , are plotted versus
dimensionless frequency, .omega..mu..sigma.S.sup.2, for various
values of the ratio of pickup length to current injection length
q/S, for a co-linear symmetric probe. It can be seen that the
voltage increases as the pickup points approach the current
injection points more closely, i.e. as q/S increases. Voltage
values calculated using the series representation for .nu.,
equation (B.23), are also shown for the probe with equally-spaced
probe points, q/S=1/3. To achieve agreement to within 2 percent of
values calculated using the exact solution at the highest frequency
considered, 30 terms in the series are required. As q/S increases,
yet more terms are needed.
In FIG. 7, the real and imaginary parts of the dimensionless
voltage, .lamda..sigma..nu..sup.RS/I, are plotted versus
dimensionless frequency, .omega..nu..sigma.S.sup.2, for various
values of the aspect ratio of a rectangular probe, c/(2S). Again,
the pickup voltage increases as the pickup points approach the
current injection points more closely, i.e. as c/S decreases. To
achieve agreement within 2 percent between values calculated using
the exact solution (equation (B.19)) and the series solution
(equation (B.23)) for a square-head probe (c/S=2) at the highest
frequency considered, 70 terms in the series are required.
In both FIG. 6 and FIG. 7, it is evident that below a certain
frequency, the pickup voltage is approximately real and constant.
In this low-frequency regime, the measured voltage matches that
obtained in the dc limit and equation (B.26) applies. Hence, in the
low-frequency regime, .nu. is independent of .mu., and .sigma. may
be determined independent of .mu. by adjusting the value of .sigma.
until theory matches low-frequency experimental data. Once .sigma.
is known, .mu. may be determined by fitting theory with
experimental data taken at higher frequencies. This procedure is
demonstrated in [16] in characterizing metal plates which are
somewhat thinner than the probe length [17].
Comparing results shown in FIG. 6 and FIG. 7 it can be seen that
the co-linear and rectangular probes perform more similarly as the
pickup points approach the current injection points more closely,
as is to be expected.
Calculation of .epsilon.
It can easily be shown that E.sup.C (equation (B.9)) is
conservative (.gradient..times.E.sup.C=0) and therefore does not
contribute to the integral around the closed loop from which
.epsilon. is derived (equation (B.4)). Hence, with equations (B.7)
and (B.10),
.times..function..function..times.d.times. ##EQU00036## Considering
the form of E.sup.W (equation (B.8)) evaluation of the integral in
equation (B.27) is straightforward, yielding
.times..times..pi..times.I.times..times..omega..times..times..mu..times..-
times..times..function..times..times..times. ##EQU00037## The
self-inductance of the pickup circuit, L, may therefore be
expressed as
.times..times..pi..times..mu..times..times..times..function..times..times-
..times..times. ##EQU00038## Complex Voltage V Combining results
(B.18) and (B.28) in accordance with equation (B.2) gives,
finally,
.times..times..pi..times..times..sigma..function..function..function..fun-
ction..function..times. ##EQU00039## where
.function..times..function..rho.eI.times..times..times..times..rho..rho.I-
.times..times..function.I.times..times..mu..times..times..times..rho..func-
tion.I.times..times..rho..times. ##EQU00040## and
.mu..sub.r=.mu./.mu..sub.0 is the relative permeability of the
half-space.
In FIG. 8 the effect of varying l on V is shown in the case of a
co-linear, symmetric probe with equally-spaced probe points
(q/S=1/3). Only the imaginary part of V is shown since .epsilon. is
purely imaginary and has no influence on the real part of V. It can
be seen that, as l increases, Im(V) becomes linear in frequency due
to the dominance of |.epsilon.| over |Im(.nu.)|. From a practical
point of view, it is important to minimize l so that the component
of V which carries information about the specimen, .nu., is not
swamped by the inductive term, .epsilon..
Experiment
The theoretical expression for the complex voltage (equations
(B.30) and (B.31)) has been validated by comparison with
experimental data. Two different four-point probes, one with
co-linear arrangement of the probe points and one rectangular, were
used. The probes were constructed by mounting four sprung, point
contacts in a plastic support block. The separation of the contacts
was measured using digital calipers. With reference to FIG. 5, the
dimensions of the probe are listed in Table 3. The uncertainty in
the dimensions derived primarily from some lateral play in the pin
position which can occur as the springs are compressed.
Measurements of complex voltage were made with the probes in
contact with a thick, alloy 2024 aluminum bloc, whose parameters
are listed in table 4. The conductivity of the block was measured
independently using an eddy-current coil. Details of the
conductivity measurement and further details of the experimental
procedure for the ACPD measurements can be found in [16, 18].
TABLE-US-00002 TABLE 3 Probe parameters. l (mm) Configuration S
(mm) p (mm) q (mm) c (mm) (fitted value) Co-linear 20.03 .+-. -17.5
.+-. 17.6 .+-. 0 2.98 .+-. 0.01 0.07 0.2 0.4 Rectangular 17.64 .+-.
-17.47 .+-. 17.55 .+-. 2.5 .+-. 0.2 2.15 .+-. 0.01 0.07 0.07
0.07
TABLE-US-00003 TABLE 4 Half-space parameters conductivity, .sigma.,
thickness, T and lateral dimensions, w .times. d. Metal Alloy
.sigma. (MSm.sup.-1) T (mm) w .times. d (mm) Aluminum 2024 17.6
.+-. 0.2 101 149 .times. 202
The dimensions of the aluminum block, with respect to the
dimensions of the probes, are such that some discrepancy between
theory and experiment due to edge effects is expected. For the
co-linear probe placed centrally on the largest face of the
aluminum block, the error due to edge effects is minimized by
orienting the line of the probe so that it is parallel with the
shorter side of the block face (w=149 mm) [10]. The error is also
reduced by employing a probe in which the four points are not
equally-spaced, but in which the pickup points are closer to the
current injection points. In fact, for the co-linear probe used in
this experiment, (q-p)/(2S).apprxeq.0.88 and w/(2S).apprxeq.3.7 w.
For these ratios, and assuming that the aluminum block is
`infinite` in the direction perpendicular to the line of the probe
(dimension d), edge effects are expected to lead to a discrepancy
of approximately 2 percent between theory and experiment in the dc
limit. Since in practice this block is finite in the direction
perpendicular to the line of the probe (d=202 mm), a discrepancy a
little larger than 2 percent is expected between theory and
experiment in the dc limit, becoming smaller as frequency
increases, due to greater confinement of the electric field in the
region of the probe. According to calculations of DCPD as a
function of the ratio of plate thickness to probe dimension (here
T/S=5[17], the thickness of the block is expected to approximate a
half-space very well, with no significant error arising due to its
finite thickness.
Experimental measurements made with co-linear and rectangular
probes are compared with theory in FIG. 9 and FIG. 10,
respectively. In both cases, there is very good agreement between
theory and experimental data. The calculated curves shown for the
imaginary part of the impedance have been obtained by adjusting the
value of the vertical dimension of the pickup circuit, l (see FIG.
5), to give the best fit to the experimental data. For the
co-linear probe, l=2.98 mm. For the rectangular probe, l=2.15 mm.
Both these values are similar to the physical values of l for these
probes. No free parameters are involved in obtaining the
theoretical curves for the real part of the impedance shown in FIG.
10 and FIG. 11. The discrepancy between theory and experiment in
the low-frequency regime is approximately 4 percent for both sets
of measurements. The fact that the measured real part of V is
larger than that predicted by theory, rather than smaller,
indicates that edge effects are likely responsible for the
discrepancy. Other significant sources of error are the uncertainty
in the probe dimensions and in the conductivity of the same.
Thus, an exact solution for the complex, frequency-dependent
voltage measured between the pickup points of a four-point probe in
contact with a metal half-space has been derived. Very good
agreement between theory and experiment on an aluminum block has
been obtained, for co-linear and rectangular arrangements of four
probe points. As well as providing a method for measuring
electrical conductivity and effective magnetic permeability of
thick metal specimens, the present invention allows for theoretical
analysis of four-point ACPD on stratified planar conductors, for
the practical purpose of nondestructive evaluation of conductive
surface treatments and coatings.
REFERENCES
The following references are cited in the disclosure, all of which
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[3] V. G. Gerasimov, A. D. Kovachev, Yu. V. Kulaev and A. D.
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